We can simulate data from a normal distribution as follows:

set.seed(144)
x1 <- rnorm(n= 1000, 0, 1)
x2 <- rnorm(n=1000, 3, 4)

data <- as.data.frame(cbind(x1, x2))
head(data)

##           x1         x2
## 1 -1.6505562  1.2206718
## 2  0.6028106 10.0211403
## 3 -0.4738839  0.9871657
## 4 -1.7976130 -0.6453392
## 5 -1.4211210  3.8133647
## 6  0.1588875  6.1606392

You could also sample form a multivariate normal distributions, but we won’t do that here.

require(MASS)
Sigma <- matrix(c(7,5,5,9),2,2)
data_mv <- as.data.frame(mvrnorm(n = 1000, rep(0, 2), Sigma))

require(DT, warn.conflicts = F)
datatable(data, options = list(pageLength = 5))

require(reshape, warn.conflicts = F)
data <- melt(data)
datatable(data, options = list(pageLength = 5))

So two beautiful normal distributions.

This doesn’t relate to the normal density stuff at all. But I need to put something here as well. So, just so you know, we can take the inverse of functions:

\[ \begin{align} g_i(\mu_{ij}) &= \nu_{i} + \lambda_i \eta_j \\ \mu_{ij} &= g_i^{-1}(\nu_{i} + \lambda_i \eta_j). \end{align} \]

These relate to the Generalized Linear Factor Analysis (GLFA) Framework. Moderated Nonlinear Factor Analysis can be seen as an extension of this GLFA (Bauer and Hussong 2009).